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Fluid Phase Equilibria 288 (2010) 67–82

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Fluid Phase Equilibria

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omparison of the prediction power of 23 generalized equations of state: Part I.aturated thermodynamic properties of 102 pure substances

. Abdollahi-Demneha,∗, M.A. Moosaviana, M.M. Montazer-Rahmatia, M.R. Omidkhahb, H. Bahmaniara

School of Chemical Engineering, College of Engineering, University of Tehran, Tehran, IranDepartment of Chemical Engineering, Tarbiat Modares University, Tehran, Iran

r t i c l e i n f o

rticle history:eceived 31 May 2009eceived in revised form8 September 2009ccepted 7 October 2009vailable online 18 November 2009

eywords:omparisonquations of state

a b s t r a c t

Accurate representation of the thermodynamic properties of pure compounds is required to increasethe robustness of an equation of state (EOS) when predicting phase equilibria for mixtures. With thispurpose, 23 generalized equations of state (EOSs) have been applied to predict the thermodynamicproperties of 102 pure substances (16,107 data points) and to report the average absolute deviationsof these properties from experimental values. Investigated thermodynamic properties are: vapor pres-sure, saturated vapor and liquid molar volume, molar enthalpy and entropy of vaporization and saturatedliquid molar isobaric heat capacity. Furthermore, behavior of the cubic and non-cubic EOSs in the vicin-ity of the critical point has been elucidated. Pure compounds which have been used in this study canbe classified as elements (monatomic and diatomic), oxides, alkanes, naphthenes, halogenated alkanes,

aturated thermodynamic propertiesaturated liquid molar isobaric heatapacities

alkenes, cyclic aliphatics, alkynes, dienes, alcohols, aromatics, ethers and miscellaneous groups (con-sisting of acetone, ethyl acetate and ammonia). Based on obtained average absolute deviations, it canbe concluded that LKP (Lee–Kesler–Plöcker et al.), TBS (Trebble–Bishnoi–Salim), TB (Trebble–Bishnoi),MNM (modified Nasrifar–Moshfeghian), MMM (Mohsen-Nia–Modarres–Mansoori), PT (Patel–Teja) andPRGGPR (Gasem–Gao–Pan–Robinson modification to the Peng–Robinson EOS) EOSs, respectively havethe most accurate predictions among the 23 studied generalized EOSs for the above-mentioned saturated

es of
thermodynamic properti
. Introduction

Generalized EOSs are widely used to represent and predicthermodynamic properties of pure components and mixtures in

any industrial applications. These kinds of equations requirepecific data such as the critical properties, the acentric factornd sometimes the dipole moment or the critical compressibilityactor of each pure species according to the three-parameter, four-arameter and five-parameter corresponding states principle. Inarticular, it has been observed that accurate prediction of pureomponent thermodynamic properties increases the robustness ofhe EOSs in predicting thermodynamic properties of mixtures inwide range of pressure and temperature conditions via appro-

riate mixing rules [1]. In other words, good agreement betweenalculated and experimental thermodynamic properties of pure
ompounds is of course not a sufficient condition for a good fittingor mixtures, but is a necessary one [2]. Process engineers need EOSss a tool to predict phase equilibrium and thermo-physical prop-rties of fluids in order to design processes and equipment. For
∗ Corresponding author. Fax: +98 21 66957784.E-mail address: [email protected] (F. Abdollahi-Demneh).

378-3812/$ – see front matter © 2009 Elsevier B.V. All rights reserved.oi:10.1016/j.fluid.2009.10.006

the 102 pure components investigated.© 2009 Elsevier B.V. All rights reserved.

example, if a process engineer decides to determine the dimen-sions of a two-phase separator, first of all he should anticipatethe composition of phases, the fraction of flow in each phase andthermo-physical properties of phases whereby he can calculateseparator dimensions. In distillation design, dependency of calcu-lations on the phase equilibrium predictions is much more severe.

During the four recent decades many EOSs have been proposedby researchers working in the field of thermodynamics. In somecases these scientists have evaluated groups of EOSs using exper-imental thermodynamic properties as a part of their investigationin order to prove the advantages of using their own EOSs [3–5]although most of them limited their comparison to RKS [2] andPR EOSs [6] only and ignored the improvements made to these twoEOSs and the ability of other proposed EOSs. Some other researchershave evaluated groups of EOS the most notable of which are men-tioned below. Trebble and Bishnoi [7] have evaluated ten cubicequations of state presented before 1987 while in their study PVTdata for 75 pure components including vapor pressure and satu-
rated vapor and liquid molar volumes have been used. In a newerresearch Maghari and Hosseinzadeh-Shahri [8] evaluated the per-formance of ten van der Waals (vdW) type equations of state topredict three regularities in dense fluids. The studied regularitieswere three out of six regularities described by Boushehri et al. [9]. A
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8 F. Abdollahi-Demneh et al. / Flu

ood review of cubic EOSs including modification to the attractionerm of vdW type EOSs, volume translated equations and applica-ion of EOSs to mixtures has been published by Valderrama [10].lso in a similar review paper, Wei and Sadus [11] studied therogress made in developing EOSs for the calculation of fluid-phasequilibria. In a valuable technical report published by IUPAC [12],eiters and de Reuck presented some golden guidelines to evaluatexisting EOSs and improve the forthcoming ones.

The objective of the first part of the present research is to exam-ne the reliability of the following 22 generalized EOSs for therediction of saturated thermodynamic properties of 102 pure sub-tances including vapor pressure, saturated vapor and liquid molarolume, molar enthalpy and entropy of vaporization and saturatediquid molar isobaric heat capacities:

RKS (Soave modification to the Redlich–Kwong (RK) EOS [13]) [2]RKTCC (Twu–Coon–Cunningham modification to the RK EOS) [14]HK (Harmens–Knapp EOS) [15]PR (Peng–Robinson EOS) [6,16]PRTCC (Twu–Coon–Cunningham modification to the PR EOS) [17]PRGGPR (Gasem–Gao–Pan–Robinson modification to the PR EOS)[18]KM (Kubic modification to the Martin EOS [3]) [19]NB (Nasrifar–Bolland EOS which is in a way a modification to theMartin EOS) [5]NM (Nasrifar–Moshfeghian EOS) [20]MNM (modified form of the NM EOS) [21]TB (Trebble–Bishnoi EOS) [4]TBS (Salim and Trebble modification to the TB EOS) [22]PT (Patel–Teja EOS) [23]PTV (Valderrama modification to the PT EOS) [24]KS (Kumar–Starling EOS) [25]MKS (Chu–Zuo–Guo modification to the KS EOS) [26]Jiuxun (totally inclusive cubic EOS) [27]MMM (Mohsen-Nia–Modarres–Mansoori EOS) [28]DPTG (Dashtizadeh–Pazouki–Taghikhani–Ghotbi EOS) [29]LKP (Plöcker et al. modification to the Lee–Kesler EOS [30]) [31]BWRSH (generalized form of Starling modification to theBenedict–Web–Rubin (BWR) EOS [32]) [33]BWRSHN (Nishumi modification to the BWRSH EOS) [34,35].

Table 1 includes formulas and parameters of the 22 EOSs men-ioned above. This table also includes the RKNB EOS which isasrifar–Bolland modification [36] to the RK EOS. In the first part ofur research (i.e. within the two-phase dome or saturation region)his EOS is the same as the RKS EOS; however in the next part of ouresearch which is dedicated to the other regions, differences willppear.

. Equations of state

EOSs are major tools for correlation and prediction of thermody-amic properties of fluids, therefore a large number of publicationseal with their development or improvement. An EOS which is validver large intervals of temperatures and pressures is a powerfulool in thermodynamic predictions. Generally, three categories ofOS can be established according to their fundamentals: empirical,heoretical and semi-empirical (or semi-theoretical). The empiri-al equations of state need a large amount of experimental data ofure components and usually contain many adjustable parameters.
heir application is restricted to a very limited number of sub-tances or just one component and their lack of predictive powereyond the pressure–temperature limit where they have beeneveloped makes them impractical for general purposes. Theoret-
cal EOSs are based on statistical thermodynamic insight whereby

se Equilibria 288 (2010) 67–82

they may represent property trends correctly even far from their fit-ting range. They have fewer parameters and these parameters havephysical meaning, however, they require time-consuming calcula-tions and suffer from limitations of existing theories making theirpredictions less accurate. Semi-empirical EOSs combine featuresof both theoretical and empirical equations, for example, they pro-vide good results for a large number of pure components and theirpredictions beyond the pressure–temperature limit of the exper-imental data used to develop them are often acceptable. Due tothese facts this is the most extensively used type of EOS for predic-tion of phase equilibrium and thermodynamic properties of fluids[12]. Moreover, semi-empirical EOSs offer the fastest way to makequantitative predictions of thermo-physical properties of pure sub-stances and mixtures with few experimental determinations usingfew adjustable parameters.

Only the last three EOSs in Table 1 are non-cubic in volume.From the mathematical point of view, because of the possibilityof analytical solution for cubic EOSs, they are more widely usedthan the non-cubic ones. Robustness of an EOS to predict proper-ties of various families of substances would be a piece of valuableinformation because knowing which EOS is the most appropriatefor a group of compounds will help process engineers to performa reliable process simulation when they encounter fluids contain-ing large proportions of components belonging to this family ofsubstances.

By applying vdW conditions at the critical point, two-parameterEOSs predict constant critical compressibility factors where thecloser this prediction is to the experimental value the better is theability of the equation to predict liquid molar volumes [6]. Thus forthis kind of EOSs the predicted critical compressibility value limitsthe range of fluids for which they are accurate. Equations havingmore than two parameters are capable of predicting the true com-pressibility factor of pure substances at the critical point. If an EOSutilizes four parameters then the fourth parameter can optimize thehardness of an EOS (via the slope of (∂p/∂v)T at elevated pressures)[4]. For a pure substance, Polishuk et al. [37] suggested approximat-ing the co-volume by the value of the liquid molar volume at thetriple point. In order to determine the temperature-dependency ofthe parameters of EOSs a target function which is often a combi-nation of deviations between EOS predictions for thermodynamicproperties and experimental values is used. The selection of thiscombination is very important and effective for improving theprediction power of an EOS. Furthermore, reduced temperaturefunctions used for determining the parameters of the EOSs aredesirable to have temperature limiting behavior, and not to havecomponent-specific parameters or switching functions because asimple generalized function facilitates a priori predictions and doesnot produce crossover anomalies in derivative properties as oftenoccurs when using switching functions.

The studied equations presented in Table 1 can be divided intosix different more general types of the TB, Martin, KS, Jiuxun, cubichard-core and truncated virial (the LKP EOS is deemed to be aBWR-type EOS). The RKS, RKTCC and RKNB EOSs use the functionalform of the RK EOS with different � functions in the attractionterm exactly in the same manner that the PRTCC and PRGGPR EOSschange the PR EOS. For the subcritical region, the RKNB equation[36] is exactly the same as the RKS EOS; however for the supercriti-cal region its � function which is based on the square-well potentialdiffers from the RKS EOS, thus as stated before in Section 1, in thesaturated region of pure components the prediction of this equa-tion is exactly the same as the RKS EOS. Peng and Robinson [6],
noting that the RKS equation predicts saturated liquid molar vol-umes poorly, changed the attractive term of the vdW EOS so thatit would predict reasonable values of hydrocarbon liquid densities.Twu et al. [14,17] used the same form of � function with differentcoefficients for subcritical and supercritical regions for the RKTCC

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Table 1Studies EOSs and their parameters.

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Table 1 (Continued )

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se Equilibria 288 (2010) 67–82 71

and PRTCC EOSs. They also maintained the continuity of the predic-tion of caloric properties at the critical point by making sure thatthe enthalpy and heat capacity departures are smooth at the criticalpoint. Gasem et al. [18] proposed a simple generalized � functionfully covering the subcritical and supercritical regions for the PREOS without the use of switching functions for the PRGGPR EOS.

HK is a three-parameter EOS whose parameters have not beendefined to predict the true critical compressibility factor (Zc) at thecritical point. Harmens and Knapp [15] stated that such an actionprevented proper prediction of molar volumes at low and high pres-sures hence they determined Zc in such a way as to give on averagethe best calculated molar volume isotherm at the critical temper-ature in the range of atmospheric pressure up to a Pr of about 5s.To improve the � function at low reduced temperatures these tworesearchers proposed a switching function as well as acentric fac-tor range-based parameters resulting in discontinuity problems inthe temperature derivative-dependent thermodynamic properties.Kumar and Starling [25] proposed the well-known most generaldensity-cubic EOS (KS) in which to develop analytical relationsfor the temperature-dependence of the parameters, thermody-namic properties of propane including density, vapor pressure andthe derivative of pressure with respect to density along severalisotherms were used. Kumar and Starling applied multi-propertyregression analysis to determine an optimum set of values forthe parameters of the KS EOS. Correspondence between theseresearchers and Martin to determine the most general density-cubic EOS [3,38,39] is interesting. Five parameters of this equationhave no physical meaning which could cause difficulties in formu-lating proper mixing rules. Moreover, since the parameters in theKS EOS have not been constrained by the vdW conditions at thecritical point, it fails to describe the true critical point for pure com-ponents resulting in less accurate results near the critical region.Chu et al. [26] modified the KS EOS to improve the aforementionedproblems by rearranging the form of the KS EOS to the vdW two-term form and applying the simple vdW one-fluid mixing rules tothe new parameters and also introducing some additional termsand transition functions for the original parameters of the KS EOS.The disadvantage of their correction method is the very sharp shapeof the correction function which will result in poor prediction forsome derived properties such as heat capacity in the vicinity ofthe critical region. Continuing the KS path, Jiuxun [27] proposed atotally inclusive cubic EOS based on four-parameter correspond-ing states whose repulsive and attractive terms increased fromthe vdW two-term into three thermodynamically consistent terms.This equation is readily amenable to mathematical manipulationsince it is completely decomposed into simple terms; hence mostof the usual thermodynamic relations can be easily integrated interms of elementary functions at constant temperature.

KM and NB are modified versions of the so-called Martin EOS [3].The Martin equation was developed for predicting vapor densitiesand performs well for this purpose; however, the temperature-dependence of the coefficients is not suitable for correlatingvapor–liquid equilibria. Due to the experimental data range usedto correlate its coefficients as a function of reduced temperature[19] as well as the temperature-dependence form applied therein,the KM EOS predictions at low reduced temperatures (0.5–0.6) arebad. Nasrifar and Bolland [5] used the same � function of the RKNBEOS in the NB EOS. The NM EOS is very similar to the PR EOS with anew form of temperature-dependence in the attraction term and atemperature-dependent co-volume [20]. The MNM EOS is a mod-ified version of the NM EOS improving the prediction power of
this equation for heavy hydrocarbons [21]. This equation utilizesa semi-empirical � function for the attraction term based on thesquare-well potential. The PTV EOS [24] is a modified form of thePT EOS whose parameters can be calculated more easily. The TB EOSis a four-parameter EOS utilizing the four-parameter correspond-

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ng states principle. This equation can be reduced to the RK, PR, NM,K and PT type of EOS and its parameters have been determinedased on a systematic thermodynamically consistent optimizationethod [4]. This EOS utilizes two temperature-dependent param-

ters consisting of new and simple types of reduced temperatureunctions each of which has only one acentric factor range-basedarameter. In spite of the constraints Trebble and Bishnoi exertedn the determination of the temperature-dependent co-volumeo prevent negative values of isobaric heat capacity, an inconsis-ency in thermodynamic property prediction using the TB EOSas identified by scientists at the Shell Research Center in Ams-

erdam [22]. Specifically, it was found that the TB EOS predictedegative values of the isochoric heat capacity and imaginary val-es for speed of sound at extremely low temperatures. Subsequent

nvestigations by the TB developers has led to the conclusion thatny temperature-dependence whatsoever in the co-volume termf vdW-type equations will result in anomalies in the predictedhermodynamic properties of pure fluids. In order to correct ther-

odynamic inconsistencies in isochoric heat capacity and speed ofound predictions of the TB EOS at extreme conditions, Salim andrebble modified its parameters and proposed the TBS EOS [22].

MMM and DPTG are cubic equations classified under the gen-ral hard-core EOS defined by Hartono and Mansoori [40]. This typef equation has a different functional repulsive term compared tohe vdW type EOSs. In their study, Mohsen-Nia et al. [28] statedhat molecular dynamic results show that the vdW-type repulsiveerm is accurate up to a packing factor of y = b/(4v) = 0.1 while the

MM repulsive term can predict the hard-sphere properties up topacking factor of 0.35. The first and second temperature deriva-

ives of the MMM EOS temperature-dependent co-volume are notontinuous at the critical temperature resulting in an anomalousehavior for the prediction of the thermal properties at and near theritical point. LKP, BWRSH and BWRSHN can be classified under theeneralized truncated virial-type equations. The eleven-constanttarling and Han equation [33] is an extension of the BWR EOSith temperature corrections for the parameters of the BWR equa-

ion using multi-property regression on pure component data tobtain values for required constants. The main disadvantage theWR EOS suffers from is that the large number of adjustable param-ters required makes it difficult to extend to mixtures. The LKPOS [31] is the same as the LK EOS [30] providing better mixingules to determine the parameters for mixtures, thus evaluationf the LKP EOS for pure substances can be attributed to the LKOS, as well. This EOS is non-cubic in volume based on Pitzer’sacroscopic theorem of three-parameter corresponding states [41]herein thermodynamic properties of normal and reference fluids

re expressed by a modified BWR EOS. Pitzer’s theorem consistsf a first-order perturbation around a simple fluid (acentric fac-or ω = 0) which can effectively be applied to moderately largeon-spherical molecules to correct non-idealities caused by molec-lar size and shape. Distinguished features of this type of EOSs areomplicated temperature-dependence and numerous parameterswhich must be determined by the reduction of plentiful, accu-ate pressure–volume–temperature and vapor–liquid equilibriumata) as well as the iterative solution required to obtain the volume.

It is worth mentioning that when someone uses availablehermodynamic relations derived from an EOS such as fugacityoefficient, enthalpy, entropy and molar isobaric heat capacityepartures presented somewhere even by the developer of thatOS, care should be taken due to mistakes caused by typographicalrrors. For instance, the fugacity expression (both for pure compo-
ents and species in a mixture) presented for the PT EOS by theuthors themselves [23], (∂pr/∂Tr)Vr
presented to calculate molarsobaric heat capacity departure of the LK EOS by the authors them-elves [30], and the entropy departure relation for the DPTG EOS29] similar to the analytic expressions of the fugacity coefficient


and entropy departure of the previous version of the MMM EOS[42] again presented by the authors themselves are not correct.Thus researchers are highly recommended to derive the thermo-dynamic properties themselves from the sequence given by Reid etal. [43] which seems to be easier than alternative sequences.

3. Results and discussion

In this study, first the vapor pressure has been obtained by sat-isfying the equality of the saturated phase fugacities at a giventemperature with a convergence criterion of |fv − fl| ≤ 10−4 kPa foreach EOS using a manipulated Newton–Raphson method [44], thenthe other saturated properties have been calculated, correspond-ingly. In the modified version of the Newton–Raphson technique,starting from an initial guess for X0 (in the current case X is thevapor pressure) which is the variable of the function I(X), the suc-cessive value for the variable Xk+1 at each iteration to find the rootof I(X) would be Xk+1 = Xk − �I(Xk)/I′(Xk). The value of � called thedamping factor is chosen as � = 1 when |I(Xk+1)| < |I(Xk)| and a stan-dard version of the Newton–Raphson method results, otherwisethe value of � is halved and Xk+1 is reevaluated. On the other hand,applying the Maxwell criterion, a recursive function for the vaporpressure can also be obtained which gives a chance to find the vaporpressure by a recursive solution method; however the convergencespeed and controllability of this method is not as good as the previ-ous applied method. In order to calculate the target function whichis the vapor and liquid phase fugacity difference, an initial guess ofthe vapor pressure should result in more than one root of volumefor an EOS. The best initial guess for the vapor pressure could beobtained from vapor pressure correlations, however, whenever thisdoes not work, comparing the obtained single-phase compressibil-ity factor at the estimated pressure with the critical one providesan alternative [45].

In order to calculate the generalized parameters of the EOSs,pure component properties such as the critical pressure andtemperature, the acentric factor and sometimes the critical com-pressibility or dipole moment are needed. The critical propertiesare well known for several species; however, discrepancies exist forheavy compounds or compounds that break down at temperaturesclose to the critical point. Also different values have been reportedfor acentric factors in different references, hence in this researchall aforementioned pure component properties together with idealisobaric heat capacity and vapor pressure relationships (as an initialguess to determine the vapor pressure from the studied EOS) havebeen taken from a unique reference which is a reliable one namelythe data bank of Reid et al. [43]. For those few substances which arenot included in this data bank or which have incomplete properties,the DIPPR compilation [46] has been used instead. Furthermore,the quoted experimental thermodynamic properties such as vaporpressure, saturated vapor and liquid molar volumes, enthalpy andentropy of vaporization and saturated liquid molar isobaric heatcapacity for 102 pure compounds (16,107 data points) have beentaken from three well-known open sources [47–49].

Saturated liquid heat capacities are not strong functions of tem-perature except above Tr = 0.7–0.8 [43]. Indeed, there is a shallowminimum in saturated liquid heat capacity as a function of temper-ature for many compounds which occurs below the normal boilingpoint while from this minimum up to the critical temperature, theliquid heat capacity increases with increasing temperature [43,47].Fig. 1 shows experimental isobaric heat capacities of saturatedliquid propylene as a function of reduced temperature [50,51] com-
pared to the predictions of the PR and LKP EOSs. As it can be seen,the trend of the predicted values does not conform to the trend ofexperimental saturated liquid heat capacities at low reduced tem-peratures. Furthermore, at reduced temperatures less than 0.48 ortemperatures whose corresponding vapor pressures is less than


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ig. 1. Predictions of the PR EOS for isobaric heat capacities of saturated liquidropylene as a function of reduced temperature compared to the experimental data50,51].

kPa (whichever is lower) and near the critical point, predictions oflmost all of the EOSs for the studied thermodynamic properties arenaccurate and may result in tremendous errors affecting the aver-ge absolute error of the investigated property significantly whilet temperatures even adjacent to the critical point (0.48 ≤ Tr ≤ 0.98)redictions are acceptable; moreover in the vicinity of the criti-al temperature, it is possible for the BWRSH, BWRSHN and KSOSs to anticipate a single phase rather than two phases at anyressure, hence in order to have a realistic judgment at the end ofhis study, experimental data points belonging to reduced temper-tures higher than 0.98 and less than 0.48 have been excluded fromhis investigation. Kedge and Trebble suggested [52] that adding aimple temperature-independent term to an EOS can improve thequation to fit the pure component saturation and PVT data, how-ver adding the Kedge and Trebble proposed term to a cubic EOSakes it non-cubic while the improvements are not significant.Because of the large number of substances, the average abso-

ute deviation of the investigated thermodynamic properties forach substance is not presented within the text and only the resultsor families (wherein substances are grouped) have been given inhe next section. Several types of compounds such as elementsmonatomic and diatomic), oxides, alkanes, naphthenes (cycloalka-es), halogenated alkanes, alkenes, cyclic aliphatics (cycloalkenes),lkynes, dienes, alcohols, aromatics, ethers and a miscellaneousroup (consisting of acetone, ethyl acetate and ammonia) haveeen included in this study. Since the BWRSH, the BWRSHN andhe KM EOSs are not compatible with components possessing neg-tive acentric factors far from zero, i.e. helium and n-hydrogenthe LKP EOS shows the poorest results for these components asell), these two components have been excluded. In the same man-er, hydrazine because of unacceptable predictions of the TBS and

iuxun EOSs for this substance that possesses a high critical com-ressibility factor of 0.429 (makes one of the parameters negativehile it is expected to be positive) has been excluded. Without

hese exclusions, a realistic judgment at the end of this researchould not be made.

It should be noted that most of the reported values fornthalpy and entropy and isobaric heat capacity of pure substancesn the available compilations are calculated from the so-calledemi-empirical multi-parameter reference EOSs whose numerousarameters have been determined based on plenty of experimen-al vapor pressures and liquid and vapor molar volumes. In other
ords, these properties are often not measured experimentally
nd are indirectly determined from other experimentally mea-ured properties. Furthermore, for pure substances the entropy ofaporization at a temperature is equal to the enthalpy of vapor-

se Equilibria 288 (2010) 67–82 73

ization at that temperature divided by the absolute temperaturewhere the enthalpy of vaporization can be calculated from theClausius–Clapeyron equation coupled with either experimentalthermal properties or the aforementioned reference EOSs.

As stated before, an EOS which is cubic in volume can besolved analytically, however to solve a non-linear non-cubic EOSnumerical methods should be applied. At a specified temperaturethe functional form of the BWRSH, BWRSHN and the modifiedBWR EOSs used for the simple and reference fluid of the LKPEOS is P = h(v), hence to find roots of these equations the mod-ified Newton–Raphson method with a convergence criterion of|Pcalc. − PGiven| ≤ 10−6 kPa has been used again. Whenever inflec-tions or multiple roots exit, the Newton–Raphson technique mayfail to converge and if it converges, the solution obtained willdepend on the initial guess. However this method converges rapidlywhen only one root exists. Similar to cubic EOSs, for the BWR-typeEOS, the rightmost intersection of the molar volume isotherm witha specified pressure line is to be construed as representing the vapormolar volume and the leftmost intersection as representing the liq-uid molar volume. The other solutions have no physical significanceand their presence is a source of difficulty in obtaining the desiredsolutions.

Based on the solution method described for BWR-type EOSs, inorder to solve this kind of EOS for volume at a specified pressure andtemperature, it is necessary to provide two good initial guesses forthe molar volume, one for the vapor phase and another for the liquidphase. According to the BWR-type EOS molar volume predictionsfor each isotherm, it is recommended that initial guesses for vaporand liquid phases should respectively result in pressures less thanor equal to and more than or equal to the specified pressure whilethe first derivative of pressure with respect to the molar volume atthese points is negative (lateral descending part of the curve). A liq-uid molar volume less than the predicted value from a cubic EOS likeTBS and a vapor molar volume more than the predicted value from acubic EOS like TBS while both satisfy the above-mentioned inequal-ities are good initial guesses. Moreover, for the vapor phase solutiondue to lower absolute values of the pressure derivative with respectto the molar volume which may result in an undesired solution, itis recommended to manipulate the molar volume change in eachiteration to confirm the above-mentioned two characteristics of thevapor molar volume initial guess.

The vdW conditions at the critical point have been exertedduring the determination of the parameters of all studied EOSsexcept the KS, BWRSH and BWRSHN EOSs. For the two-parametercubic EOSs studied, applying the vdW conditions without truncatednumbers will result in the constant critical compressibility fac-tors reported in the literature, however it should be noted that formost of them, applying the reported truncated numbers by the EOSdevelopers (often with six significant figures) will result in differ-ent critical compressibility factors. In order to make the calculationof EOS parameters at the critical point easier, the KM, HK and PTVEOSs utilize explicit expressions which are function of the criticalcompressibility or the acentric factor, thus they are approximatelycompatible with the vdW criteria. A comparison between the stud-ied EOSs predictions for Zc based on truncated constants reportedby the EOSs developers and non-truncated constants is presentedin Table 2.

As a general rule, if parameters of a cubic EOS are constrainedby the vdW conditions at the critical point, then for a pure com-ponent the predicted critical temperature and pressure of the EOSachieved from the contact point of the left-hand and right-hand

fying the equality of fugacity condition for the saturated phases,and the true critical temperature and pressure would be the same.Otherwise, like the KS EOS the predictions for the critical temper-ature and pressure will differ from the experimental values while

74 F. Abdollahi-Demneh et al. / Fluid Phase Equilibria 288 (2010) 67–82

Table 2Calculated critical compressibility with and without truncated constants.

EOS RKS RKTCC PR PRTCC PRGGPR NM MNM KM NB TB TBS

Zc (Non-trunc.) 0.333 0.333 0.307 0.307 0.307 0.302 0.302 I (ω) 0.329 1.075Zc 1.063Zc

Zc (Trunc.) 0.347 0.333 0.321 0.307 0.321 0.296 0.295 I (ω) 0.320 1.075Zc 1.063Zc

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EOS PT PTV Jiuxun KS MKS

Zc (Non-trunc.) I (ω) I (Zc) Zc n.a. Zc

Zc (Trunc.) I (ω) I (Zc) Zc n.a. Zc

he difference between the predicted critical compressibility factornd its experimental value depends on the number of parametersf the EOS and how they are determined. The reason behind thisule for the cubic EOS is the variation form of pressure in terms ofhe molar volume within the critical isotherm. If the general formf a pressure-explicit cubic EOS is considered as:

= k(v, T)g(v, T)

= RTv2 + av + b

v3 + cv2 + dv + e(1)

Eq. (1) gives the following cubic molar volume equation:

(P, v, T) = Pg(v, T) − k(v, T) = ˛v3 + ˇv2 + �v + ı = 0 (2)

For a pure substance, parameters a, b, c, d, e are just a functionf temperature, thus at the critical temperature Eqs. (1) and (2) cane written as:

= k(v, Tc)g(v, Tc)

= k(v)g(v)

= RTv2 + acv + bc

v3 + ccv2 + dcv + ec(3)

(P, v, Tc)=ϕ(P, v)=Pg(v) − k(v)=˛cv3+ˇcv2+�cv+ıc=0 (4)

On the other hand, at the critical pressure the first and seconderivatives of Eq. (3) with respect to the molar volume are bothero:

′ = k′(v)g(v) − k(v)g′(v)

g(v)2= 0 gives P = k(v)

g(v)= k′(v)

g′(v)(5)

′′ = k′′(v)g(v) − k(v)g′′(v)

g(v)2= 0 gives P = k(v)

g(v)= k′′(v)

g′′(v)(6)

At the critical temperature, the first and second derivatives ofpressure-explicit cubic equation of ϕ(P, v, T) with respect to theolar volume become:

′(Tc, P, v) = P ′g(v) + Pg′(v) − k′(v) (7)

′′(Tc, P, v) = P ′′g(v) + 2P ′g′(v) + Pg′′(v) − k′′(v) (8)

nserting Eqs. (5) and (6) into the two equations above at the criticalressure makes them zero, thus the cubic Eq. (4) whose first andecond derivatives at the critical pressure are zero can be writtens follows:

(Pc, v, Tc) = ˛c

(v + ˇc

3˛c

)3

(9)

q. (9) denotes the fact that at the critical point an EOS cubic inolume has only one root and this root satisfies the vdW conditions.

In the case of non-cubic EOS, the validity of this rule highlyepends on the functional form of the EOS. For instance, the LKPOS critical isotherms for the reference fluid and the simple fluidre both monotonically descending in terms of the molar volumend both satisfy the vdW conditions at the critical pressure, resem-ling Eq. (9). During the development of the BWRSH and BWRSHN
OSs the vdW conditions at the critical point have not been applied,ence for a pure component their related critical isotherm form isften not desirable and the predicted critical temperature and pres-ure of these EOSs differ from the true values resulting in anomalousehavior in the vicinity of the critical point.
MMM DPTG HK LKP BWRSH BWRSHN

0.308 0.333 I (ω) I (ω) n.a. n.a.0.299 0.316 I (ω) I (ω) n.a. n.a.

Tables 3–8 show the average absolute percentage deviation ofvapor pressure, saturated liquid molar volume, saturated vapormolar volume, molar enthalpy of vaporization, molar entropy ofvaporization and saturated liquid molar isobaric heat capacity pre-dictions of the studied EOSs for different families of compounds.Table 9 presents the substances included in each family, the numberof pure component data points used for each investigated propertyand the reduced temperature range. In this study, data which havebeen used for the evaluation of the vapor pressure, liquid molarvolume, vapor molar volume, latent heat of vaporization, latententropy of vaporization and saturated isobaric heat capacity arerespectively with uncertainties of ±0.02, ±0.016, ±0.017, ±0.18,±0.18 and ±1% in the subcritical region while in the vicinity of thecritical point uncertainties respectively reach ±0.006, ±0.4, ±0.8,±0.22, ±0.22 and ±5% [53–55]. On the other hand, to have an overalljudgment at the end a unique uncertainty is needed for each studiedthermodynamic property therefore values of ±0.02, ±0.02, ±0.2,±0.2 and ±1% are respectively supposed to be overall uncertaintiesof the vapor pressure, vapor and liquid molar volumes, latent heatof vaporization, latent entropy of vaporization and saturated iso-baric heat capacity data. Before final judgment about the studiedEOSs, the abovementioned experimental data uncertainties havebeen added to the relevant average absolute percentage deviationsreported in Tables 3–8.

From saturated vapor pressure deviation results presented inTable 3 coupled with the relevant experimental data uncertain-ties, it can be concluded that the most accurate predictions (forthe investigated data points) for the different groups presentedin Table 3, according to the given group sequence belong tothe MNM/PR/PRTCC, TB/PR, MNM, MMM/LKP, PT, PRGGPR, TB,LKP/PRTCC, DPTG, RKS/NB, LKP/TBS/RKTCC, TB/PTV and HK EOSs,respectively. Overall, the least average absolute deviations forvapor pressure predictions belong to the PRTCC, RKTCC, MNM, PT,PR, PRGGPR, MMM, LKP and TBS EOSs with deviations between1 and 2%, while the RKS, NB, HK, DPTG and Jiuxun EOSs also havesmall errors between 2 and 3%. Almost all studied EOSs have accept-able predictions of vapor pressure for the studied pure substances;however vapor pressure predictions of the NM and BWRSH EOSsfor those components possessing high acentric factors (e.g. alkanesheavier than nonane) are poor since heavy hydrocarbons data havenot been used during the generalization procedure for obtainingtheir parameters.

From saturated liquid molar volume deviation results shownin Table 4 coupled with the relevant experimental data uncer-tainties, it can be concluded that the most accurate predictions(for the investigated data points) for different groups presentedin Table 4 according to the given group sequence belong to theKS/MKS, Jiuxun, LKP, KS, Jiuxun, LKP, PT, Jiuxun, KS, Jiuxun, BWR-SHN, KS and Jiuxun EOSs, respectively. Overall, the least averageabsolute deviations for liquid molar volume predictions belong tothe Jiuxun and KS EOSs with deviations between 1 and 2%, the MKS,
BWRSHN, BWRSH, TBS and TB EOSs with deviations between 2 and3% and LKP and PTV with deviations between 3 and 4%. Based onthe results obtained, it is suggested not to use RK-type EOSs for theprediction of liquid density of pure substances.

F. Abdollahi-Demneh et al. / Fluid Phase Equilibria 288 (2010) 67–82 75

Table 3aAverage absolute percentage deviation of vapor pressure predictions of the studied EOSs for pure substances.

Group RKS RKTCC PR PRTCC PRGGPR NM MNM KM NB TB TBS

Elements 2.25 1.08 1.06 1.07 1.43 1.35 1.05 3.68 1.62 3.70 2.21Oxides 3.02 2.55 2.13 2.54 2.24 3.65 2.75 4.86 2.70 2.11 2.89Alkanes 1.69 1.20 1.48 1.21 1.47 7.94 1.15 4.29 1.96 2.13 1.84Cycloalkanes 1.75 1.55 2.08 1.57 2.11 1.66 1.64 4.86 1.94 1.71 1.60Halogenated alkanes 2.05 1.75 1.81 1.72 1.95 1.89 1.76 3.17 2.14 3.23 1.81Alkenes 2.22 1.58 1.61 1.55 1.42 2.43 1.66 5.89 1.99 4.96 1.69Cycloalkenes 2.99 2.85 3.16 2.84 2.94 2.69 3.19 10.54 2.78 2.10 3.46Alkynes 3.09 2.78 3.37 2.72 3.11 2.74 2.76 6.57 2.89 3.61 2.76Dienes 2.79 2.78 2.95 2.79 3.08 2.63 2.10 3.27 3.23 2.72 2.55Alcohols 4.08 4.12 4.81 4.21 4.57 12.89 4.47 9.95 4.09 8.70 9.26Aromatics 1.56 1.08 1.35 1.10 1.34 2.70 1.12 4.27 1.52 3.47 1.07Ethers 2.70 2.57 2.42 2.72 2.99 3.82 2.02 2.07 3.23 1.16 1.87Miscellaneous group 1.55 1.44 1.45 1.53 1.85 1.82 1.36 2.61 1.86 3.02 1.65

Total 2.05 1.64 1.78 1.63 1.84 3.69 1.64 4.19 2.10 3.13 1.99

Table 3bAverage absolute percentage deviation of vapor pressure predictions of the studied EOSs for pure substances.

Group PT PTV Jiuxun KS MKS MMM DPTG HK LKP BWRSH BWRSHN


Total 1.72 3.09 2.96 3.65 3.88 1.96 2.37 2.13 1.98 6.55 4.07

Table 4aAverage absolute percentage deviation of saturated liquid molar volume predictions of the studied EOSs for pure substances.



Total 13.94 13.86 7.16 6.98 6.98 8.56 4.56 6.64 9.25 2.81 2.77

Table 4bAverage absolute percentage deviation of saturated liquid molar volume predictions of the studied EOSs for pure substances.



Total 5.08 3.86 1.74 1.97 2.05 5.00 7.32 5.80 3.74 2.47 2.34

76 F. Abdollahi-Demneh et al. / Fluid Phase Equilibria 288 (2010) 67–82

Table 5aAverage absolute percentage deviation of saturated vapor molar volume predictions of the studied EOSs for pure substances.


Elements 2.72 1.44 2.04 1.90 2.47 2.18 2.23 4.52 2.03 3.92 3.43Oxides 5.37 5.20 3.86 4.25 3.81 5.64 4.85 7.47 4.43 3.22 4.93Alkanes 1.85 1.63 1.96 1.63 1.78 1.94 2.06 3.54 1.77 2.63 2.43Cycloalkanes – – – – – – – – – – –Halogenated alkanes 2.42 2.34 2.43 2.10 2.12 2.35 2.63 4.04 2.01 3.68 2.60Alkenes 1.52 1.28 1.14 0.81 0.82 1.03 1.12 4.78 0.47 3.96 1.65Cycloalkenes – – – – – – – – – – –Alkynes 2.15 2.48 1.67 1.40 1.25 1.73 2.25 1.86 1.08 2.30 2.53Dienes 1.19 0.58 1.60 1.76 1.96 1.80 1.41 2.73 1.36 3.28 3.05Alcohols 12.12 12.22 11.52 11.75 11.48 13.21 12.21 10.89 12.06 10.62 11.19Aromatics 6.44 6.40 7.13 6.44 6.90 6.85 6.68 9.11 6.20 10.28 7.14Ethers 5.02 5.22 4.86 4.42 4.34 3.82 5.65 5.14 4.32 7.03 5.40Miscellaneous group 3.37 3.48 3.00 2.48 2.40 2.64 3.80 4.31 2.64 4.47 2.64

Total 2.85 2.67 2.73 2.45 2.51 2.79 2.96 4.52 2.43 3.90 3.09

Table 5bAverage absolute percentage deviation of saturated vapor molar volume predictions of the studied EOSs for pure substances.


Elements 1.74 4.88 4.24 3.13 3.17 2.38 1.84 2.71 2.46 2.79 2.90Oxides 4.14 5.53 6.11 6.20 8.14 4.33 8.40 5.18 4.98 5.90 6.46Alkanes 1.75 2.85 4.30 3.36 3.09 2.01 3.13 2.24 1.78 3.23 3.64Cycloalkanes – – – – – – – – – – –Halogenated alkanes 2.40 3.49 4.59 4.55 4.56 2.43 3.93 2.95 2.33 4.49 4.99Alkenes 0.65 2.69 3.37 1.40 1.47 1.10 2.58 1.85 1.87 1.26 1.71Cycloalkenes – – – – – – – – – – –Alkynes 1.41 2.25 3.82 1.07 0.87 1.51 4.42 2.13 2.50 1.57 2.25Dienes 1.24 1.37 6.48 1.55 1.37 2.53 2.26 2.04 0.51 2.17 1.98Alcohols 11.86 13.69 11.46 14.92 14.31 10.92 14.92 13.27 11.17 16.94 12.56Aromatics 6.83 7.72 19.28 8.26 8.30 6.74 7.54 6.77 6.53 7.34 8.30Ethers 5.56 6.31 5.21 4.61 4.28 4.14 7.99 6.32 4.44 3.17 4.38Miscellaneous group 3.14 8.06 4.40 6.86 23.78 2.06 5.52 3.53 3.36 8.01 6.14

Total 2.64 4.10 5.21 4.52 5.39 2.71 4.29 3.27 2.74 4.49 4.72

Table 6aAverage absolute percentage deviation of �Hvap predictions of the studied EOSs for pure substances.



Total 2.75 2.33 2.30 2.23 2.18 2.77 2.20 4.11 2.63 2.25 2.30

Table 6bAverage absolute percentage deviation of �Hvap predictions of the studied EOSs for pure substances.



Total 2.33 2.58 4.97 2.63 3.79 2.43 3.07 2.41 1.90 2.80 2.75

F. Abdollahi-Demneh et al. / Fluid Phase Equilibria 288 (2010) 67–82 77

Table 7aAverage absolute percentage deviation of �Svap predictions of the studied EOSs for pure substances.



Total 2.85 2.43 2.42 2.33 2.30 2.86 2.31 4.22 2.74 2.36 2.41

Table 7bAverage absolute percentage deviation of �Svap predictions of the studied EOSs for pure substances.



Total 2.44 2.70 5.05 2.76 3.81 2.52 3.18 2.52 2.03 2.91 2.86

Table 8aAverage absolute percentage deviation of saturated liquid molar isobaric heat capacity predictions of the studied EOSs for pure substances.


Elements 12.82 12.67 10.68 11.03 10.60 8.16 14.75 49.64 12.68 9.06 12.75Oxides 21.85 15.55 18.11 13.97 19.90 33.19 24.85 36.35 22.35 14.41 17.53Alkanes 7.83 11.47 8.28 9.83 6.42 7.61 6.99 24.08 7.80 6.48 6.39Cycloalkanes 5.43 10.31 8.37 9.33 3.26 7.25 7.58 45.97 5.11 5.49 3.74Halogenated alkanes 7.85 9.46 6.95 7.92 6.26 9.87 7.00 17.71 8.01 5.62 6.61Alkenes 7.34 13.04 9.10 11.16 5.85 6.59 5.75 36.03 7.19 5.91 5.31Cycloalkenes – – – – – – – – – – –Alkynes – – – – – – – – – – –Dienes 2.17 9.94 4.21 7.81 0.96 5.70 6.24 44.55 2.09 4.40 1.75Alcohols 26.90 13.99 22.93 15.46 25.37 41.72 29.47 18.03 27.64 20.36 23.21Aromatics 4.88 8.88 5.95 7.71 4.21 6.70 4.97 13.79 4.78 4.30 3.81Ethers – – – – – – – – – – –Miscellaneous group 11.30 10.85 7.53 8.38 9.38 16.64 11.13 25.91 11.74 5.12 8.60

Total 9.22 10.97 8.60 9.38 7.62 10.8 8.91 25.31 9.30 6.71 7.68

Table 8bAverage absolute percentage deviation of saturated liquid molar isobaric heat capacity predictions of the studied EOSs for pure substances.


Elements 11.77 12.55 13.80 10.81 33.36 9.10 9.55 12.37 5.34 10.84 8.76Oxides 18.23 16.59 8.73 22.59 237.15 21.91 30.01 18.91 10.99 25.62 23.96Alkanes 8.06 8.04 6.36 7.76 42.42 6.35 8.26 7.43 3.39 9.81 7.83Cycloalkanes 7.72 8.29 6.54 16.75 16.75 0.90 9.97 4.33 1.22 4.77 2.76Halogenated alkanes 6.95 6.75 6.82 6.66 44.03 6.01 8.11 6.47 3.62 7.42 6.19Alkenes 8.45 8.67 12.95 9.76 32.45 6.17 7.10 6.95 3.32 7.61 6.30Cycloalkenes – – – – – – – – – – –Alkynes – – – – – – – – – – –Dienes 3.32 3.90 4.11 13.45 13.45 0.99 10.55 1.41 2.74 1.42 0.54Alcohols 21.30 20.11 11.79 29.58 31.05 22.27 34.66 23.30 22.50 24.95 27.85Aromatics 5.82 5.97 6.19 7.65 7.81 4.08 6.96 4.60 3.05 5.01 3.86Ethers – – – – – – – – – – –Miscellaneous group 7.93 7.44 9.85 11.91 102.60 9.52 14.20 9.22 6.18 5.66 6.05

Total 8.54 8.43 8.30 9.25 51.88 7.44 10.0 8.12 4.46 9.15 7.83

78F.A

bdollahi-Dem

nehet

al./FluidPhase

Equilibria288

(2010)67–82

Table 9Number of data points, reduced temperature range and reference of experimental properties of pure substances used in this study.

Compound VP data vl data vv data �H data �S data Cpl data Reference

No. Tr range No. Tr range No. Tr range No. Tr range No. Tr range No. Tr range

ElementsNeon 10 0.55–0.95 10 0.55–0.95 10 0.55–0.95 10 0.55–0.95 10 0.55–0.95 10 0.55–0.95 [47]Argon 14 0.56–0.96 14 0.56–0.96 14 0.56–0.96 14 0.56–0.96 14 0.56–0.96 14 0.56–0.96 [47]Krypton 11 0.55–0.96 11 0.55–0.96 11 0.55–0.96 11 0.55–0.96 11 0.55–0.96 11 0.55–0.96 [47]Xenon 13 0.56–0.97 13 0.56–0.97 13 0.56–0.97 13 0.56–0.97 13 0.56–0.97 13 0.56–0.97 [47]Nitrogen 29 0.50–0.95 29 0.50–0.95 29 0.50–0.95 29 0.50–0.95 29 0.50–0.95 29 0.50–0.95 [48]Oxygen 17 0.48–0.97 17 0.48–0.97 17 0.48–0.97 17 0.48–0.97 17 0.48–0.97 17 0.48–0.97 [49]Fluorine 15 0.48–0.97 15 0.48–0.97 15 0.48–0.97 15 0.48–0.97 15 0.48–0.97 15 0.48–0.97 [47]Chlorine 19 0.53–0.97 19 0.53–0.97 19 0.53–0.97 19 0.53–0.97 19 0.53–0.97 18 0.53–0.94 [47]Bromine 15 0.48–0.95 15 0.48–0.95 15 0.48–0.95 15 0.48–0.95 15 0.48–0.95 15 0.48–0.95 [47]

OxidesWater 40 0.49–0.96 40 0.49–0.96 40 0.49–0.96 40 0.49–0.96 40 0.49–0.96 40 0.49–0.96 [47]Carbon monoxide 24 0.51–0.95 24 0.51–0.95 24 0.51–0.95 24 0.51–0.95 24 0.51–0.95 0 – [48]Carbon dioxide 39 0.71–0.98 39 0.71–0.98 39 0.71–0.98 39 0.71–0.98 39 0.71–0.98 27 0.90–0.98 [48]Nitrous oxide 12 0.60–0.98 12 0.60–0.98 12 0.60–0.98 12 0.60–0.98 12 0.60–0.98 0 – [47]Sulfur dioxide 22 0.49–0.97 22 0.49–0.97 22 0.49–0.97 22 0.49–0.97 22 0.49–0.97 20 0.49–0.93 [47]

AlkanesMethane 72 0.48–0.97 72 0.48–0.97 72 0.48–0.97 72 0.48–0.97 72 0.48–0.97 20 0.48–0.97 [47,49]Ethane 59 0.49–0.98 59 0.49–0.98 59 0.49–0.98 59 0.49–0.98 59 0.49–0.98 16 0.49–0.98 [47,49]Propane 79 0.49–0.98 79 0.49–0.98 79 0.49–0.98 79 0.49–0.98 79 0.49–0.98 79 0.49–0.98 [47,49]n-Butane 79 0.49–0.98 79 0.49–0.98 79 0.49–0.98 79 0.49–0.98 79 0.49–0.98 79 0.49–0.98 [47,49]i-Butane 75 0.48–0.98 75 0.48–0.98 75 0.48–0.98 75 0.48–0.98 75 0.48–0.98 75 0.48–0.98 [47,49]n-Pentane 25 0.50–0.99 26 0.50–0.99 18 0.64–0.99 9 0.54–0.67 9 0.54–0.67 7 0.50–0.61 [48]i-Pentane 23 0.50–0.98 25 0.50–0.98 20 0.60–0.98 9 0.50–0.68 9 0.50–0.68 9 0.48–0.65 [48]n-Hexane 26 0.50–0.97 27 0.50–0.97 24 0.54–0.97 22 0.54–0.97 22 0.54–0.97 6 0.49–0.59 [48]n-Heptane 28 0.50–0.99 27 0.50–0.99 28 0.50–0.99 27 0.50–0.99 27 0.50–0.99 16 0.50–0.78 [47,48]n-Octane 28 0.51–0.97 27 0.51–0.97 26 0.53–0.97 27 0.52–0.97 27 0.52–0.97 4 0.51–0.54 [48]i-Octane 27 0.50–0.98 27 0.50–0.98 27 0.50–0.98 27 0.50–0.98 27 0.50–0.98 7 0.50–0.57 [48]n-Nonane 25 0.50–0.98 21 0.50–0.98 15 0.50–0.98 29 0.50–0.98 29 0.50–0.98 4 0.50–0.54 [47,48]n-Decane 16 0.52–0.97 16 0.52–0.97 16 0.52–0.97 16 0.52–0.97 16 0.52–0.97 1 0.518 [47,48]n-Undecane 17 0.54–0.79 0 0 – 1 NBP 1 NBP 0 – [48]n-Dodecane 17 0.55–0.80 13 0.55–0.73 0 – 1 NBP 1 NBP 0 – [48]n-Tridecane 14 0.56–0.76 13 0.56–0.76 0 – 1 NBP 1 NBP 0 – [48]n-Tetradecane 15 0.57–0.77 14 0.57–0.76 0 – 1 NBP 1 NBP 0 – [48]n-Pentadecane 15 0.58–0.78 14 0.58–0.77 0 – 1 NBP 1 NBP 0 – [48]n-Hexadecane 10 0.64–0.77 13 0.60–0.77 0 – 1 NBP 1 NBP 0 – [48]n-Heptadecane 16 0.59–0.80 15 0.59–0.78 0 – 1 NBP 1 NBP 0 – [48]n-Octadecane 15 0.60–0.79 13 0.60–0.77 0 – 1 NBP 1 NBP 0 – [48]n-Nonadecane 15 0.61–0.81 13 0.61–0.77 0 – 1 NBP 1 NBP 0 – [48]n-Eichosane 16 0.61–0.81 12 0.61–0.76 0 – 1 NBP 1 NBP 0 – [48]

Naphthenes (cycloalkanes)Cyclo pentane 27 0.50–0.98 7 0.51–0.61 0 – 26 0.50–0.98 26 0.50–0.98 6 0.49–0.59 [48]Methyl cyclopentane 25 0.49–0.96 7 0.49–0.59 0 – 19 0.49–0.94 19 0.49–0.94 5 0.49–0.56 [48]Ethyl cyclopentane 30 0.48–0.99 5 0.50–0.55 0 – 29 0.50–0.99 29 0.50–0.99 0 – [48]1,1-Dimethylcyclopentane 18 0.50–0.79 5 0.51–0.57 0 – 27 0.50–0.97 27 0.50–0.97 0 – [48]Cyclohexane 28 0.51–0.98 9 0.51–0.64 0 – 27 0.51–0.98 27 0.51–0.98 3 0.50–0.53 [48]Methyl cyclohexane 30 0.49–0.98 29 0.49–0.98 0 – 29 0.49–0.98 29 0.49–0.98 3 0.49–0.52 [48]Ethyl cyclohexane 18 0.48–0.76 0 – 0 – 31 0.48–0.97 31 0.48–0.97 4 0.48–0.51 [48]

F.Abdollahi-D

emneh

etal./Fluid

PhaseEquilibria

288(2010)

67–8279

Halogenated alkanesCarbon tetrachloride 27 0.50–0.97 27 0.50–0.97 27 0.50–0.97 27 0.50–0.97 27 0.50–0.97 25 0.50–0.93 [47]Carbon tetrafluoride 12 0.48–0.97 12 0.48–0.97 12 0.48–0.97 12 0.48–0.97 12 0.48–0.97 12 0.48–0.97 [47]Chloroform 13 0.52–0.97 13 0.52–0.97 13 0.52–0.97 13 0.52–0.97 13 0.52–0.97 8 0.67–0.93 [47]Methyl chloride 34 0.50–0.98 34 0.50–0.98 34 0.50–0.98 34 0.50–0.98 34 0.50–0.98 23 0.50–0.77 [47]Difluoro methane 65 0.61–0.97 65 0.61–0.97 65 0.61–0.97 65 0.61–0.97 65 0.61–0.97 0 – [49]Trichlorofluoro methane 59 0.49–0.98 59 0.49–0.98 59 0.49–0.98 59 0.49–0.98 59 0.49–0.98 58 0.49–0.97 [49]Dichlorodifluoro methane 66 0.50–0.98 66 0.50–0.98 66 0.50–0.98 66 0.50–0.98 66 0.50–0.98 66 0.50–0.98 [49]Chlorotrifluoro methane 64 0.57–0.98 64 0.57–0.98 64 0.57–0.98 64 0.57–0.98 64 0.57–0.98 36 0.67–0.90 [49]Bromotrifluoro methane 18 0.50–0.97 18 0.50–0.97 18 0.50–0.97 18 0.50–0.97 18 0.50–0.97 18 0.50–0.97 [47]Chlorodifluoro methane 62 0.52–0.97 62 0.52–0.97 62 0.52–0.97 62 0.52–0.97 62 0.52–0.97 62 0.52–0.97 [49]Trifluorom ethane 61 0.58–0.97 61 0.58–0.97 61 0.58–0.97 61 0.58–0.97 61 0.58–0.97 4 0.77–0.87 [47,49]Trifluoro ethane 18 0.49–0.95 18 0.49–0.95 18 0.49–0.95 18 0.49–0.95 18 0.49–0.95 18 0.49–0.95 [47]Trifluorotrichloro ethane 67 0.50–0.98 67 0.50–0.98 67 0.50–0.98 67 0.50–0.98 67 0.50–0.98 53 0.56–0.93 [49]Dichlorotetrafluoro ethane 67 0.52–0.97 67 0.52–0.97 67 0.52–0.97 67 0.52–0.97 67 0.52–0.97 58 0.56–0.94 [49]Chlorodifluoro ethane 73 0.54–0.97 73 0.54–0.97 73 0.54–0.97 73 0.54–0.97 73 0.54–0.97 39 0.68–0.87 [49]Difluoro ethane 66 0.50–0.98 66 0.50–0.98 66 0.50–0.98 66 0.50–0.98 66 0.50–0.98 62 0.58–0.96 [49]Dichlorofluoro methane 18 0.54–0.97 18 0.54–0.97 18 0.54–0.97 18 0.54–0.97 18 0.54–0.97 0 – [49]Dichlorofluoro ethane 72 0.53–0.88 72 0.53–0.88 72 0.53–0.88 72 0.53–0.88 72 0.53–0.88 0 – [49]Tetrafluoro ethane 66 0.49–0.98 66 0.49–0.98 66 0.49–0.98 66 0.49–0.98 66 0.49–0.98 66 0.49–0.98 [49]Pentafluoro ethane 67 0.60–0.98 67 0.60–0.98 67 0.60–0.98 67 0.60–0.98 67 0.60–0.98 57 0.66–0.98 [49]Chlorotetrafluoro ethane 66 0.54–0.98 66 0.54–0.98 66 0.54–0.98 66 0.54–0.98 66 0.54–0.98 66 0.54–0.98 [49]Chloropentafluoro ethane 21 0.63–0.97 21 0.63–0.97 21 0.63–0.97 21 0.63–0.97 21 0.63–0.97 0 – [49]

AlkenesEthylene 68 0.49–0.98 68 0.49–0.98 68 0.49–0.98 68 0.49–0.98 68 0.49–0.98 67 0.49–0.97 [49]Propylene 71 0.49–0.98 71 0.49–0.98 71 0.49–0.98 71 0.49–0.98 71 0.49–0.98 71 0.49–0.98 [49]1-Butene 23 0.48–0.98 10 0.48–0.70 0 – 22 0.48–0.98 22 0.48–0.98 7 0.48–0.60 [48]cis-2-Butene 14 0.49–0.76 7 0.51–0.65 0 – 22 0.49–0.97 22 0.49–0.97 7 0.48–0.62 [48]trans-2-Butene 12 0.52–0.78 7 0.52–0.66 0 – 20 0.52–0.96 20 0.52–0.96 6 0.49–0.60 [48]i-Butylene 22 0.49–0.96 9 0.49–0.68 0 – 21 0.49–0.96 21 0.49–0.96 7 0.48–0.61 [48]1-Pentene 25 0.48–0.97 4 0.61–0.65 0 – 24 0.48–0.97 24 0.48–0.97 10 0.48–0.65 [48]cis-2-Pentene 17 0.49–0.80 11 0.49–0.68 0 – 24 0.49–0.97 24 0.49–0.97 9 0.49–0.63 [48]trans-2-Pentene 17 0.49–0.81 11 0.49–0.68 0 – 24 0.49–0.98 24 0.49–0.98 9 0.49–0.63 [48]1-Hexene 17 0.48–0.80 12 0.54–0.94 0 – 26 0.48–0.98 26 0.48–0.98 0 – [48]1-Heptene 17 0.51–0.81 5 0.52–0.58 0 – 26 0.50–0.97 26 0.50–0.97 4 0.50–0.56 [48]1-Octene 18 0.50–0.80 7 0.50–0.59 0 – 27 0.50–0.97 27 0.50–0.97 0 – [48]

Cyclic Aliphatics (Cycloalkenes)Cyclopentene 16 0.48–0.78 5 0.56–0.62 0 – 26 0.48–0.97 26 0.48–0.97 0 – [48]Cyclohexene 16 0.49–0.75 5 0.51–0.56 0 – 28 0.49–0.97 28 0.49–0.97 0 – [48]

AlkynesAcetylene 13 0.62–0.97 13 0.62–0.97 13 0.62–0.97 13 0.62–0.97 13 0.62–0.97 0 – [47]Methyl acetylene 16 0.48–0.85 5 0.53–0.63 0 – 21 0.48–0.98 21 0.48–0.98 0 – [48]

Dienes1,3-Butadiene 45 0.52–0.98 45 0.52–0.98 45 0.52–0.98 45 0.52–0.98 45 0.52–0.98 9 0.52–0.71 [48]2-Methyl-1,3-butadiene 15 0.48–0.77 5 0.59–0.65 0 – 24 0.48–0.96 24 0.48–0.96 0 – [48]

AlcoholsMethanol 20 0.56–0.97 20 0.56–0.97 20 0.56–0.97 20 0.56–0.97 20 0.56–0.97 19 0.56–0.95 [47]Ethanol 16 0.68–0.97 16 0.68–0.97 16 0.68–0.97 16 0.68–0.97 16 0.68–0.97 11 0.68–0.88 [47]1-Propanol 14 0.54–0.97 22 0.51–0.97 21 0.55–0.97 25 0.53–0.97 25 0.53–0.97 0 – [48]2-Propanol 14 0.54–0.97 6 0.54–0.79 0 – 7 0.61–0.73 7 0.61–0.73 0 – [48]

Aromatic hydrocarbonsBenzene 27 0.52–0.98 27 0.52–0.98 27 0.52–0.98 27 0.52–0.98 27 0.52–0.98 25 0.52–0.94 [47]Toluene 21 0.49–0.98 21 0.49–0.98 21 0.49–0.98 21 0.49–0.98 21 0.49–0.98 20 0.49–0.95 [47]

80 F. Abdollahi-Demneh et al. / Fluid Pha

Tabl

e9

(Con

tinu

ed)

Com

pou

nd

VP

dat

av l

dat

av v

dat

a�

Hd

ata

�S

dat

aC

pld

ata

Ref

eren

ce

No.

T rra

nge

No.

T rra

nge

No.

T rra

nge

No.

T rra

nge

No.

T rra

nge

No.

T rra

nge

m-X

ylen

e34

0.51

–0.9

80

–0

–30

0.51

–0.9

830

0.51

–0.9

81

–[4

8]o-

Xyl

ene

310.

50–0

.97

0–

0–

310.

50–0

.97

310.

50–0

.97

1–

[48]

p-X

ylen

e30

0.51

–0.9

80

–0

–30

0.51

–0.9

830

0.51

–0.9

89

0.51

–0.7

7[4

8]Et

hyl

ben

zen

e19

0.51

–0.8

011

0.51

–0.6

70

–30

0.51

–0.9

830

0.51

–0.9

81

–[4

8]Is

opro

pyl

ben

zen

e20

0.50

–0.8

06

0.50

–0.5

80

–31

0.50

–0.9

731

0.50

–0.9

73

0.51

–0.5

9[4

8]D

iph

enyl

200.

49–0

.96

200.

49–0

.96

200.

49–0

.96

200.

49–0

.96

200.

49–0

.96

190.

49–0

.94

[47]

Nap

hth

alen

e28

0.50

–0.8

611

0.66

–0.7

913

0.66

–0.8

237

0.50

–0.9

837

0.50

–0.9

80

–[4

8]

Eth

ers

Eth

ylet

her

210.

56–0

.98

200.

59–0

.98

200.

59–0

.98

160.

65–0

.98

160.

65–0

.98

0–

[48]

Phen

ylet

her

130.

68–0

.84

150.

64–0

.83

110.

70–0

.83

120.

7–0.

8412

0.7–

0.84

0–

[48]

Mis

cell

aneo

us

grou

pA

ceto

ne

210.

59–0

.96

210.

59–0

.96

210.

59–0

.96

200.

59–0

.94

200.

59–0

.94

170.

65–0

.94

[47]

Eth

ylac

etat

e25

0.52

–0.9

825

0.52

–0.9

825

0.52

–0.9

817

0.67

–0.9

817

0.67

–0.9

80

–[4

8]A

mm

onia

710.

48–0

.97

710.

48–0

.97

710.

48–0

.97

710.

48–0

.97

710.

48–0

.97

670.

57–0

.97

[49]

Tota

l31

3727

1623

8330

8430

8417

03


According to the saturated vapor molar volume deviation resultsgiven in Table 5 coupled with the relevant experimental data uncer-tainties, it can be concluded that the most accurate predictions(for the investigated data points) for different groups presented inTable 5 according to the given group sequence belong to the RKTCC,TB, PRTCC/RKTCC, n.a., NB, NB, n.a., MKS, LKP, TB, NB, BWRSHand MMM EOSs, respectively. Overall, the least average absolutedeviations for vapor molar volume predictions belong to the NB,PRTCC, PRGGPR, PT, RKTCC, MMM, PR, LKP, NM, RKS and MNMEOSs, respectively with deviations from 2.4 to 3% while the othershave average absolute deviations less than 5.4%. Due to the fact thatthe saturated vapor molar volume varies significantly with tem-perature or pressure, accurate prediction of vapor molar volume isinterrelated with vapor pressure prediction thus, except for alco-hols, all studied EOSs, similar to the vapor pressure results, showacceptable predictions for the saturated vapor molar volumes ofthe studied pure substances.

Based on the molar enthalpy of vaporization deviation resultspresented in Table 6 coupled with the relevant experimental datauncertainties, it can be concluded that the most accurate predic-tions (for the investigated data points) for the different groupspresented in Table 6 according to the given group sequence belongto the DPTG/RKTCC/LKP/PRTCC, TB, LKP, RKS/NB/RKTCC/HK, LKP,PRGGPR/TBS/LKP/PRTCC/PT/MNM/PR, NB/TBS/NM/MMM, TB/LKP,LKP, NB/PRTCC/RKS/PRGGPR, KS, LKP and TBS EOSs, respectively.It is perceived from the deviations presented within Table 7 thatexcept for dienes, alcohols and aromatics for which the MKS,NB/PRTCC/RKS/PRGGPR/Jiuxun and KS/LKP EOSs, respectively havethe most accurate predictions for the molar entropy of vaporiza-tion, the same conclusion as was drawn for the molar enthalpy ofvaporization is valid for entropy as well. Overall, the least aver-age absolute deviations for the molar enthalpy and entropy ofvaporization belong to the LKP EOS with deviations of 1.9 ± 0.2and 2.0 ± 0.2%, respectively while the other EOSs have deviationsbetween 2 and 3%, except the DPTG and MKS EOSs with deviationsbetween 3 and 4% and the KM and Jiuxun EOSs with deviationsbetween 4 and 5%.

Reported results within Table 8 for the saturated liquid molarisobaric heat capacity coupled with the relevant experimental datauncertainties imply that the most accurate predictions (for theinvestigated data points) for the different groups presented inTable 8 according to the given group sequence belong to the LKP,Jiuxun, LKP, LKP/MMM, LKP, LKP, BWRSHN/PRGGPR/HK/BWRSH,Jiuxun, LKP/TBS/BWRSHN, n.a. and TB/BWRSH/BWRSHN EOSs,respectively. Overall, the least average absolute deviations for theliquid heat capacity predictions belong to the LKP and TB EOSs withdeviations of 4.5 ± 1 and 6.7 ± 1% while the MMM, PRGGPR, TBS andBWRSHN EOSs have deviations between 7 and 8%. The worst over-all predictions belong to the MKS EOS with 51.9 ± 1 and the KMEOS with 25.1 ± 1 average absolute percentage deviations and theremaining equations have deviations between 8 and 11%.

It was observed that for pure components almost all EOSs havepoor predictions for the second derivative-based properties suchas heat capacity in the reduced temperature range of 0.9–1. Forexample, it was observed that predictions of the BWRSH and BWR-SHN EOSs for the saturated liquid molar isobaric heat capacity ofn-butane at reduced temperatures near the critical point (Tr ≥ 0.95)are really far from the experimental values and the predictions ofthe other EOSs. This tremendous difference persuaded us to explorethe behavior of the BWRSH EOS within this region. Fig. 2 illustratesthe variation of the isobaric heat capacity in terms of the reduced
temperature for normal butane at different constant reduced pres-sures. This figure consists of two parts; part (a) allocated to theliquid phase and part (b) belonging to the vapor phase. As it canbe seen from the reduced isobar of 0.9174 approaching the satura-tion point which separates the liquid and vapor phases, the isobaric


Fbv

hctpbi(ipat(c

Ft

ig. 2. The BWRSH EOS predictions for molar isobaric heat capacity isobars of n-utane: (a) liquid phase from Tr = 0.5174 up to saturation reduced temperature; (b)apor phase from saturation reduced temperature up to Tr = 1.

eat capacity reaches higher values. It should be noted that the heatapacity relies on the first and second temperature derivatives ofemperature-dependent expressions of EOSs, thus the more com-lex these functionalities are, the more likely the anomalies in theehavior of this property would be. Based on the vdW conditions,

f temperature and pressure approach the critical conditions, then∂p/∂�)v should approach zero resulting in an infinite heat capac-ty; thus the heat capacity predictions of an EOS around the criticaloint highly depend on the behavior of (∂p/∂�) in this region and
vccordingly on the functional form of the EOS as well as the fit-ed parameters. For the BWRSH and BWRSHN EOSs lower values of∂p/∂�)v in the critical region cause higher values of heat capacitiesompared with experimental values.
ig. 3. Average absolute percentage deviation of the KM EOS predictions for inves-igated thermodynamic properties of methane in terms of reduced temperature.

se Equilibria 288 (2010) 67–82 81

The KM EOS at very low reduced temperatures (and to someextent at reduced temperatures near the critical point) has itsworst predictions especially for the saturated liquid molar iso-baric heat capacity due to the complex temperature-dependenceof its parameters. This can be seen from Fig. 3 which illustratesthe KM EOS average absolute deviations of the investigated sat-urated thermodynamic properties of methane in terms of thereduced temperature. As stated before, the MKS equation becauseof the sharp behavior of the transition exponential function usedto change its parameters behaves erroneously in the reduced tem-perature region of 0.95 <Tr <1.05 where the transition functionbecomes non-zero and multiplying by the second temperaturederivatives of the temperature-dependent terms leads to the worstresults for the saturated liquid specific heat capacities among allEOSs.

4. Conclusions

The prediction power of 23 generalized EOSs has been evalu-ated for 102 pure compounds. The accuracy of predictions of mostof the studied EOSs is good and approximately in the same range(differences are not significant) for all thermodynamic propertiesexcept for the saturated liquid molar volume and the saturatedliquid molar isobaric heat capacity. The saturated liquid molarvolume and saturated liquid molar isobaric heat capacity predic-tions of a few of the studied EOSs is good and the most accurateresults belong to the Jiuxun and LKP EOSs, respectively. Basedon the obtained average absolute deviations of the investigatedthermodynamic properties for the studied 102 pure substances,it is concluded that LKP, TBS, TB, MNM, MMM, PT and PRGGPREOSs provide the most accurate predictions among all studiedEOSs.

As a general rule, applying the vdW conditions at the criticalpoint to the cubic EOSs will make them predict the true criticaltemperature and pressure at the contact point of the vapor andliquid saturation lines while the difference between the predictedcritical compressibility factor and its experimental value dependson the number of parameters of the EOS and how they are deter-mined. For non-cubic EOS the above-mentioned rule depends onthe functional form of the critical isotherm.

List of symbolsb co-volume (m3/kgmol)Cv molar isochoric heat capacity (kJ/(kgmol K))Cp molar isobaric heat capacity (kJ/(kgmol K))f fugacity (kPa)g general cubic function in Eq. (1)h functional form of EOS (kPa)H molar enthalpy (kJ/kgmol)I typical functionk general quadratic function in Eq. (1)n.a. not applicableNBP normal boiling pointP pressure (kPa)R gas constant (8.314 kJ/(kgmol K))S molar entropy (kJ/(kgmol K))T temperature (K)v molar volume (m3/kgmol)X a general variableZ compressibility factor

Greek letters˛, ˇ, � , ı parameters of cubic Eq. (2)� damping factor� molar density (kgmol/m3)ω Pitzer’s acentric factor

8 id Pha

ϕ

ScCGklrv0

S′′

A

aoocoa

R

[[[[[[

[

[[

[[[[[[[[[[

[

[[

[[[[[[[[[[

[[

[[

[

[

[

[

[


general cubic function in Eq. (2)difference between vapor and liquid phase property

ubscriptscritical conditions

alc. calculated from an EOSiven given to an EOS

iteration numberliquid phasereduced conditionsvapor phaseinitial guess

ubscriptsfirst derivative

′ second derivative

cknowledgements

The authors dedicate this research to Esmaeel Alizadeh who hadn effective role in the inception of this research. He left us becausef an accident but we never forget him and his beautiful smile. Onef the authors, F. Abdollahi-Demneh, acknowledges the Iran LNGompany for moral as well as financial support during this stagef the project. The authors thank Mohammad-Amin Nasri for hisssistance in preparing the data bank.

eferences

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Documents

Comparison of the prediction power of 23 generalized ...lllee/K10DemnehEOS.pdf · triple point. In order to determine the temperature-dependency of the parameters of EOSs a target